[tex]f(x)=e^{x-11}-8\to y=e^{x-11}-8\ \ \ \ |add\ 8\ to\ both\ sides\\\\e^{x-11}=y+8\\\\\ln e^{x-11}=\ln(y+8)\iff x-11=\ln(y+8)\ \ \ \ |add\ 11\ to\ both\ sides\\\\x=\ln(y+8)+11\\\\a)\ \boxed{f^{-1}(x)=\ln(x+8)+11}\\\\b)\ The\ domain\ of\ f^{-1}(x):\\\\x+8 > 0\to \boxed{x > -8\to x\in(-8;\ \infty)}[/tex]
